论文标题
分数硬质操作员的泊松问题：分配身份和单数解决方案
The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions
论文作者
论文摘要
本文的目的是研究和分类泊松问题的奇异解决方案$$％\ begin {equation} \ label {eq 0.1} \ left \ left \ {\ begin {aligned} {\ mathcal l} \ {0 \}，\\ u = 0 \ Quad \ {\ rm in} \ \，{\ Mathbb r}^n \ setMinusω％\\％\ liminf_ \ liminf_ {x \ \ x \ \ x \ 0} /φ_μ（x）= k。 \ end {Aligned} \ right。 $$用于分数强的操作员$ {\ MATHCAL L}_μ^s U =（-Δ）这里$（δ）^s $，$ s \ in（0,1）$，是订单$ 2S $的分数laplacian和$μ\geμ__0$，其中$μ_0= -2^{2S} \ frac {γ^2（\ frac {n+2s} 4）}} {γ^2（\ frac {n -2s} {4} {4}} <0 $是分数hardy hardy notquality in Bractional hardy norquality in t ractional hardy norquality。该分析需要对基本解决方案和相关的分布身份进行详尽的研究。将特别注意关键案例$μ=μ_0$，该$比$ use>μ_0$需要更多的微妙估计。
The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_μu = f \quad\ {\rm in}\ \, Ω\setminus \{0\},\\ u =0 \quad\ {\rm in}\ \, {\mathbb R}^N \setminus Ω %\\ %\liminf_{x \to 0}\:|u(x)| /Φ_μ(x) = k. \end{aligned} \right. $$ for the fractional Hardy operator ${\mathcal L}_μ^s u= (-Δ)^s u +\fracμ{|x|^{2s}}u$ in a bounded domain $Ω\subset {\mathbb R}^N$ ($N \ge 2$) containing the origin. Here $(-Δ)^s$, $s\in(0,1)$, is the fractional Laplacian of order $2s$, and $μ\ge μ_0$, where $μ_0 = -2^{2s}\frac{Γ^2(\frac{N+2s}4)}{Γ^2(\frac{N-2s}{4})}<0$ is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case $μ= μ_0$ which requires more subtle estimates than the case $μ>μ_0$.
